Analytical results for the capacitance of a circular plate capacitor

Abstract

We study the classic problem of the capacitance of a circular parallel plate capacitor. At small separations between the plates, it was initially considered in the 19th century by Kirchhoff, who found the leading and the subleading term in the capacitance. Despite a large interest in the problem, almost 150 years later, only the second subleading term has been found analytically. Using the recent advances in the asymptotic analysis of Fredholm integral equations of the second kind with finite support, here we study the one governing the circular capacitor, known as the Love equation. We find analytically many subleading terms in the capacitance at small separations. We also calculate the asymptotic expansion at large separations, thus providing two simple expressions which practically describe the capacitance at all distances. The approach described here could be used to find exact analytical expansions for the capacitance to an arbitrary number of terms in regimes of both small and large separations.